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Journal of Vibration Testing and System Dynamics 8(3) (2024) 309--316 | DOI:10.5890/JVTSD.2024.09.003

Jonathan N. Blakely$^1$, Marko S. Milosavljevic$^1$, Ned J. Corron$^1$, Seth D. Cohen$^2$, Casey Fendley$^2$

$^1$ U.S. Army DEVCOM Aviation & Missile Center, Redstone Arsenal, Alabama 35898, USA

$^2$ Kratos SRE, Birmingham, Alabama 35211, USA

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Abstract

The properties of nonlinear dynamics and chaos are shown to be fundamental to optimal communication signals subject to two practical and realistic design requirements: (i) operation in a noisy environment and (ii) simple hardware implementation. The first requirement implies the receiver should include a matched filter, i.e. a filter that maximizes the signal-to-noise ratio when receiving the corresponding matched waveform. The second requirement can be met by employing a simple infinite-impulse-response (IIR) filter as the matched filter. Here we examine the waveforms matched to stable IIR filters characterized by all-pole transfer functions. This class of filters contains many of the most popular and widely used filter families, including Butterworth, Chebyshev (type I), and Bessel. We find that these waveforms are chaotic in the sense that they are deterministic and characterized by a positive Lyapunov exponent. Interestingly, a return map using samples from any such waveform takes the form of a shift map. We derive this map theoretically and present an experimental reconstruction of it using an actual electronic filter and its matched waveform. A practical consequence of chaos in these waveforms is the potential for simple and efficient signal generation using chaotic oscillators.

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